TRAJECTORIES AND DISTRIBUTION OF INTERSTELLAR DUST GRAINS IN THE HELIOSPHERE

A big step toward constraining MHD models of the heliosphere was taken when the Voyager 1 and Voyager 2 spacecraft encountered the termination shock at different distances. These data, together with the ∼78 offset between the upwind interstellar H⁰ and He⁰ directions (the so-called hydrogen deflection plane; Lallement et al. 2005, 2010), and the 48° angle between the He⁰ upwind direction and ISMF given by the center of the IBEX ribbon, have established the asymmetry of the global heliosphere. The magnetically distorted heliosphere is confirmed by IBEX energetic neutral atom (ENA) maps showing that the tail of the heliosphere is offset from the downwind gas direction by ∼40° (Schwadron et al. 2011), and by ENA fluxes from the polar regions that reveal that the inner heliosheath in the north is thicker by ∼27 AU than in the south (Reisenfeld et al. 2012).

The heliosphere models that we use in our dust trajectory calculations, described more fully by Pogorelov et al. (2008) and Heerikhuisen et al. (2006, 2008), are mixed magnetohydrodynamic/kinetic models with the ions treated as a fluid while neutrals are treated kinetically using a statistical approach, with both particle species coupled self-consistently by charge exchange. These models have been successful in explaining the location of the IBEX ribbon of energetic neutral atoms (which is centered ∼17° away from the interstellar field direction in our models; McComas et al. 2009) and the north–south asymmetry of the heliosphere, and are consistent with the offset between the inflows of H⁰ and He⁰ into the heliosphere. The coordinate system used in the calculations is centered on the Sun and has the z-axis pointed toward the ecliptic longitude of the upwind direction, i.e., ecliptic longitude λ = 259°, but in the ecliptic plane, ecliptic latitude β = 0, which makes it point about 5° away from the upwind direction. The x-axis is perpendicular to the ecliptic plane, and the y-axis, completing the right-handed coordinate system, lies in the ecliptic plane. The particular model we use in this work has an initial field direction in the undisturbed ISM of (λ, β) = (61°, −30°) (ecliptic coordinates, see Table 1 for the direction in galactic coordinates). The projection of the B field onto the x–z plane can be seen in Figure 1 where we compare the two heliosphere models that we use with different SWMF polarities. As the interstellar plasma encounters the heliosphere, the field is draped around it, creating a curvature that tends to divert the small grains around the heliosphere. As discussed below, the grains tend to be positively charged by the solar UV radiation once they get within ∼100 AU of the Sun.

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The solar wind in the heliosphere models has a radial velocity at 1 AU of 450 km s⁻¹ and density of 7.4 cm⁻³ and has a Parker spiral magnetic field with strength 37.5 μG. These boundary conditions are then advected to the inner edge of the 3D grid at R = 10 AU. This results in values in the innermost parcel of grid of v_r ≈ 454 km s⁻¹, n_p ≈ 0.07 cm⁻³, and B_ϕ ≈ 3.3 μG (near the ecliptic). The temperature evolution in the wind is strongly affected by the interactions between interstellar neutrals and the solar wind ions that produce the PUI, and is anisotropic at 10 AU with values ranging from 2800 K to 14,000 K with a mean of 3800 K. These and other boundary conditions in the heliosphere models are summarized in Table 1. It has long been known that the large-scale solar magnetic field changes polarity every solar cycle (Howard 1977, and references therein) and these polarity changes propagate outward with the solar wind. We explore models with both different polarities for the SWMF in this work, the focusing polarity, which has north pole negative (i.e., north ecliptic pole is south magnetic pole, a.k.a. the A⁻ configuration) and the de-focusing polarity, which has north pole positive (A⁺ configuration). The reason that the field with north pole negative polarity is focusing for the dust is that, as we discuss further below, the grains are positively charged. Thus, $q \textbf {{\em v}}\ \times \textbf {\em B}$ points down toward the ecliptic for grains in the north (i.e., above the ecliptic plane) and points up toward the ecliptic for grains in the south. Note that the models we use are steady flow models and so the solar wind magnetic polarity does not evolve. The ISM before the encounter with the heliosphere is assumed to be flowing with speed 26.4 km s⁻¹ relative to the Sun and to have a total proton density of 0.06 cm⁻³ and temperature of 6527 K. The neutral H density in the ISM is assumed to be 0.15 cm⁻³.

The heliosphere models use a 3D grid in spherical coordinates (with the polar axis directed toward the longitude of the heliosphere nose), with an inner boundary (radius) at 10 AU and an outer boundary at 1200 AU with 224 zones in the r-direction (logarithmically spaced), 144 zones in θ, and 83 in ϕ. The heliosphere model results include the electron temperature, proton density, magnetic field, and velocity over this grid. The grain density grid used is a 3D Cartesian grid with a spacing of 5 AU in each direction and extending from −400 AU to +400 AU in each dimension. Note that the computation of the grain trajectories starts farther upwind (900 AU) but the densities are only recorded once the grains enter the density grid. Inside of 10 AU, the dynamical variables are assumed to have their values at the inner edge of the grid.

We focus on compact olivine-type silicate grains in our models in this paper. Slavin & Frisch (2008) have shown that absorption line data toward CMa provide strong evidence that C is not depleted from the gas phase in the LIC, which appears to be the source of the gas flowing into the solar system (Frisch et al. 2011). The depletion pattern of refractory elements in the LIC suggests a grain composition similar to that of olivines, which appear to be widespread in the local ISM (Frisch et al. 2011). Kimura et al. (2003a) conclude that C in the local ISM is slightly depleted, but exclude the low H i column density lines of sight in this determination. Thus, their method includes mostly gas that is not in the LIC but rather in other clouds. The differences between silicate and carbonaceous grains for the purposes of this paper include the size-to-mass ratio (i.e., the density of the grain material), the charging properties, and the grain optical properties (which help determine the value of β, see below). If future work indicates C depletion in the LIC or the presence of carbonaceous interstellar grains in the heliosphere, a study of such grains would be warranted, though we do not include them in the present study. The solid material density of the grains that we use is 3.3 g cm⁻³, as appropriate for olivine silicates, in the results presented in this paper. It has been suggested (e.g., Mathis 1996) that some or even most interstellar grains may have a porous or fluffy structure leading to a considerably lower material density. The first fluffy grain models have been ruled out, however, based on the excessive amount of infrared emission they would produce (Dwek 1997). It is currently unclear what fraction of interstellar grains might be fluffy. We intend to explore alternative grain types in future work.

The charging properties of interstellar grains are still quite uncertain. In our calculations, we use the optical constants and photoelectric yields of Weingartner et al. (2006), making use of their code (kindly provided to us by J. Weingartner), though we have made small modifications to include the effect of gas–grain motion on the charging (using methods described in Guillet et al. 2007). The main sources of charging for interstellar grains in the heliosphere are electron impacts and sticking and photoelectric charging by the FUV radiation field. In regions of hot gas (T ≳ 10⁵ K), electrons have enough energy to cause the ejection of secondary electrons, which tends to positively charge the grains. The resultant grain potentials that we find for the upstream direction toward the heliosphere nose are illustrated in Figure 2. The enhanced grain charging in the inner heliosheath, originally discussed by Kimura & Mann (1999), is clearly visible. We find that the grain potential is strongly position and grain size dependent. While the FUV field is dominated by solar radiation in the inner heliosphere, the background interstellar radiation field dominates in the farther regions of the outer heliosphere (see Figure 3). We use the UV radiation field of Gondhalekar et al. (1980) for the background interstellar field (e.g., see the radiation longward of 912 Å in Figure 1 of Slavin & Frisch 2008). This field corresponds to a UV flux, averaged over grain photoelectric yield, of 0.74 times that of Draine & Salpeter (1979). For the solar field, we use data from the Solar EUV Experiment (SEE) on the NASA TIMED (Thermosphere Ionosphere Mesosphere Energetics and Dynamics; http://lasp.colorado.edu/see/l3_data_page.html) mission which provides daily EUV/FUV spectra based on solar observations and models and from the SORCE (the Solar Radiation & Climate Experiment; http://lasp.colorado.edu/sorce/index.htm) mission that provides better visible/near UV data.

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For the inclusion of the effects of radiation pressure and gravity we calculate β, the ratio of the radiation pressure force to the gravitational force as a function of grain size,

$$\begin{equation} \beta = -\frac{F_r}{F_g} = \frac{R^2}{G M_\odot m_\mathrm{gr}} \frac{1}{c} \int F_\lambda \sigma _\mathrm{gr} Q_\mathrm{p}\,d\lambda , \end{equation} \tag{ 1 }$$

where F_r is the radiation pressure force, F_g is the gravitational force, R is the distance from the Sun, m_gr is the grain mass, F_λ is the flux (erg cm⁻² s⁻¹ Å⁻¹) at the distance R, σ_gr is the physical cross section of the grain, and Q_p is the radiation pressure efficiency factor. (For more details, see discussions in Gustafson 1994 and Sterken et al. 2012.) Note that R drops out since the flux goes as 1/R². This calculation depends on the grain optical properties, through Q_p, which in turn depends on the absorption efficiency, Q_a, and scattering efficiency, Q_s,

$$\begin{equation} Q_\mathrm{p} = Q_\mathrm{a} + (1 - g) Q_\mathrm{s}, \end{equation} \tag{ 2 }$$

where g ≡ 〈cos (θ)〉 is the scattering asymmetry factor. We calculate the radiation pressure efficiency using results and code from Draine (2003). Using the solar spectrum and our assumed grain density, ρ_gr = 3.3 g cm⁻³, we find that β is less than one for all grain sizes. We note that β is sensitive to the grain density and composition (Figure 2 in Kimura et al. 2003b), and β exceeds 1 (net repulsion of grains) for low-density (fluffy) grains. There is some observational evidence for repulsion of grains via radiation pressure, primarily for small grains within 4 AU of the Sun (Figure 2 in Mann 2010). Sterken et al. (2012) model variations in β and show that β is important only within several AU of the Sun (except for downstream where the effects extend farther), in accord with the observations. Given our assumptions, however, the net effect from radiation pressure and gravity on the grains is gravitational focusing though the effect is minor throughout most of the heliosphere. The peak value for β is 0.89 at a_grain ≈ 0.2 μm and so gravity and radiation pressure nearly balance and grains near that size experience little effect from the Sun's gravity or radiation pressure. As we discuss below and in Slavin et al. (2010), in our models, smaller grains are mostly excluded from the inner heliosphere by the SWMF, so that uncertainties in β do not strongly affect those results. Since the value of β decreases as 1/a_grain above 0.2 μm, the gravitational effects on the grain trajectories are greatest for the largest grains. These grains also have the smallest charge-to-mass ratios, so gravitational effects play a dominant role in determining their trajectories near the Sun.

The primary outputs of our calculations are the 3D dust density distributions. In Figures 4–6, we show slices of the dust density relative to the ambient value for the z–y, z–x, and y–x planes, respectively, comparing results for the de-focusing SWMF orientation which has magnetic north (top row) with those for the focusing SWMF polarity (bottom row). These figures are for the smallest grain sizes calculated, a_grain = 0.01, 0.0178, and 0.0316 μm. In Figures 7–9, we show the same type of plots for larger grains, a_grain = 0.0562, 0.1, and 0.178 μm and in Figures 10–12 for the largest grain sizes calculated, a_grain = 0.316, 0.562, and 1.0 μm. The Sun-centered coordinate system used is described above (Section 2.1). Here, the plots are oriented such that the heliosphere is viewed from above for the z–y plots, where the upwind direction is to the right and ecliptic longitude increases in negative y-directions. The z–x plots show a meridian cut. The y–x plots are looking downstream from a viewpoint in the upstream ISM.

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An abundance of features are apparent in these model data, but a few stand out immediately. First, it is clear that for the de-focusing SWMF polarity only fairly large grains, a_gr ≳ 0.3 μm, can reach Earth's orbit, though grains as small as 0.1 μm can penetrate the termination shock. Thus, it is hard to reconcile this model with the observed size distribution, which includes grains as small as ∼0.05 μm. For the focusing polarity field, again, grains as small as 0.1 μm can penetrate the termination shock, but in this case the focusing effects of the SWMF allows such grains to reach all the way to the Sun, as they are concentrated in the ecliptic plane. Indeed even grains as small as 0.02 μm can reach the Sun for the focusing polarity. These strong differences in the distributions of dust grains of a given size between the different magnetic polarities persists, even up to 1 μm grains, though for larger grains the focusing/de-focusing effects of the SWMF are relatively small and mitigated by the effects of gravitational focusing by the Sun. Even for the largest grains, a = 1 μm, the dust focusing tail shows some solar cycle dependence (e.g., Figures 10–12). For both SWMF polarities, the dust trajectories and thus dust density distributions are quite sensitive to grain size. This is because the gyroradii of the grains range from <1 AU for 0.01 μm grains to ∼10⁵ AU for 1 μm grains at the heliopause.

In the dust density figures, it is clear that the smallest grains are effectively completely excluded from the inner heliosheath, being diverted along with the interstellar plasma around the heliopause. At larger grain sizes, more penetration of the heliopause and the effects of the larger gyroradii can be seen. As an example, in Figure 5 in the upper right panel (a = 0.0316 μm), the low densities for large x values above the heliopause show how grains that have been diverted do not follow the heliopause, but gain a substantial +x velocity that carries them away from the heliosphere. The regions with significant dust density inside the heliopause in this panel are from scattering of grains in from the flanks of the heliopause (because their gyroradii are large enough to allow them to cross the heliopause). The enhancement in grain density in the region just upwind of the heliopause in the z–y plot for this same grain size is also caused by the decoupling. The effects of decoupling of the grains from the gas become much more pronounced for larger grain sizes. For 0.0562 μm grains, the effects of scattering off the heliopause are clear in Figures 7–9. In the z slices in particular, it can be seen that the grains, which have a small initial −x velocity (recall that the upwind direction is ∼5° above the +z-axis) are deflected by about 200 AU over the ∼150 AU distance since encountering the heliopause. We note that this grain size is peculiar in our results because for both the de-focusing and focusing SWMF models we find a zero density for the voxels near the Sun, although for the focusing SWMF smaller grains (0.0178 and 0.0316 μm) do produce non-zero density. This result, which we discuss more below, appears to be caused by the fact that grains of this size have gyroradii at the location of encountering the heliopause, that are roughly the same size as the heliopause. They thus decouple substantially from the gas, but are strongly diverted from their inflow paths.

Smaller grains, while more tightly coupled to the plasma (via the magnetic field), can scatter and leak into the heliosphere wherein they may be focused toward the ecliptic for the focusing SWMF. Larger grains, while even more decoupled from the plasma, are less diverted from their paths and, again for the focusing SWMF, have an enhanced dust density near the Sun. As the grain size increase, grains in the range of ∼0.05–0.2 μm (Figures 7–12) show increasing degrees of penetration of the heliopause and are focused either near the ecliptic poles, for the de-focusing SWMF, or in the ecliptic, for the focusing SWMF. The 3D grain density distribution for 0.178μm grains is illustrated in Figure 13, which shows a rendering of the 3D surfaces corresponding to dust density enhancements (relative to the interstellar density) of 2, 2.75, 3.5, 4.25, and 5. Going to the largest grain sizes (Figures 10–12), the grains are less and less affected by the heliopause and at the largest size, 1.0 μm, are primarily affected by the gravitational focusing by the Sun.

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The individual grain trajectories can be quite complex, especially for the intermediate grain sizes, ∼0.05–0.3 μm, for which the grains are significantly diverted from their original path but have large enough gyroradii to penetrate to the inner heliosphere. The scattering of the grains inside the heliosphere is clear as even grains that start out on similar trajectories diverge strongly from each other. This scattering is primarily caused by the fact that the gyroradii are as large as or larger than a typical length scale for variation in the magnetic field direction so the field sampled by different grains on their trajectories is substantially different and becomes more so as they propagate closer to the heliopause and inner heliosphere. The differences in trajectories are also amplified by the increase in magnetic field strength caused by compression in the heliosheath.

One of the most important reasons for calculating grain trajectories through the heliosphere starting from the undisturbed ISM is to allow us to infer the initial interstellar grain size distribution given the size distribution observed by the Ulysses and Galileo spacecrafts. With accurate 3D heliosphere models and grain trajectory calculations one could, in principle, infer from the observations the interstellar size distribution, though of course grain sizes that are either completely excluded from the inner heliosphere or not observable because of instrumental limitations cannot be constrained by this method. In this study, we have made progress toward this goal by employing accurate, though non-evolving, heliosphere models. As we discuss further below, however, the lack of solar cycle evolution of the heliosphere, in particular the SWMF, during the time the grains traverse the inner heliosphere limits the applicability of our results. Assuming either a focusing or de-focusing SWMF during an entire grain trajectory effectively exaggerates the focusing or de-focusing the grains will experience. We expect that the truth will lie in between these extreme cases. The detailed behavior of the grains with small to moderate gyroradii as the SWMF evolves is unclear at present especially considering the complex magnetic topology caused by the evolution and propagation of the heliospheric current sheets (Nerney et al. 1995). Despite these limitations, we find below that comparisons between our models and the in situ data provide valuable insights into the properties of the grains and their interaction with the heliosphere.

Our calculations do not assume any particular size distribution (being carried out independently for each grain size) since the grains are not assumed to interact significantly with each other. This assumption should be excellent over the length scales relevant for the heliosphere, though in interstellar shocks (with length scales hundreds of thousands of times longer), grain–grain collisions are a major source of dust destruction (Jones et al. 1994). The output of our calculations provide an unbiased result for each grain size, since the dust density at each point in space is calculated relative to its value in the ISM prior to interaction with the heliosphere.

Comparisons between the models and the in situ observations are not useful for the smallest grains or for regions of the model that are infrequently populated with grains. For the smallest grains, below the detection threshold of the Ulysses and Galileo dust detectors (m ≈ 7 × 10⁻¹⁶ g or a ≈ 0.04 μm for our assumed grain density), we obviously have no information on their abundance in the heliosphere. Use of our theoretical data is also limited by our statistics: if we have no counts in a voxel, this simply gives us an upper limit on our predicted dust density relative to the initial interstellar dust density for grains in that size range. In the calculations that we present in this paper, we expect, on average, ∼62 counts per voxel (5 AU in each dimension) if the dust density were undisturbed from its value in the ambient ISM. Thus, if the dust density for a particular grain size is reduced by a factor of ≳ 62 then we are likely to get no counts in the voxel and we are not sensitive to how large the reduction in dust density is above that threshold.

Although our calculations use a regular grid of starting points for the trajectories, they are, in some sense, similar to Monte Carlo calculations. Each individual trajectory is quite sensitive to its initial conditions, so the starting positions, which have no a priori justification other than being sufficiently far upstream of the heliosphere, could be considered to be random choices. A slight shifting of the starting grid would lead to considerably different individual trajectories, though the dust density results would be essentially unchanged. In addition, as mentioned above, the initial gyrovelocity for each grain has a random direction other than that is in the plane perpendicular to the magnetic field (i.e., 90° pitch angle). This small addition of randomness to the initial velocities is sufficient to remove most of the identifiable signature of the initial grid in the final dust density results. Thus, we can treat the dust density results in a statistical manner and assume the uncertainty in the "true" number of counts we should get for each voxel is consistent with Poisson statistics. As shown by Gehrels (1986), for small numbers of counts the usual $\sqrt{N}$ approximation is not a good approximation for the confidence limits and we use instead his results to determine the statistical 1σ error bars. The 1σ upper limits are $1 + N + \sqrt{N + ({3}/{4})}$ and lower limits are $N (1 - ({1}/{9N}) - ({1}/{3\sqrt{N}}))^3$. Thus, for a voxel in with no counts, the upper limit is $1 \,{+}\,\sqrt{{3}/{4}}$.

To scale the results of our models to actual densities in the ISM or heliosphere, we need to assume an initial grain size distribution including the scaling relative to the gas density. Differences between the initial grain distribution at "infinity" and the observed in situ dust distribution should then be explained in principle by grain propagation models. In Figure 14, we show the data from Frisch et al. (2009, and references therein) for Ulysses and Galileo dust detection along with models and inferred grain size distributions. Here, rather than the grain size distribution, dn/da, we plot the grain mass per logarithmic mass bin as was done in Frisch et al. (1999) and Frisch et al. (2009). Plotted this way, it is apparent that most of the mass is in grains on the large end of the scale. In the figure, we plot the standard MRN power-law distribution for grain sizes (Mathis et al. 1977, dashed, purple curve) which extends from 0.005 μm to 0.25 μm with the normalization set by the assumption of an interstellar density of n_H = 0.22 cm⁻³ and a gas-to-dust mass ratio of 150. The H density and gas-to-dust ratio are taken from the LIC photoionization models of Slavin & Frisch (2008), which are successful in that they predict interstellar boundary conditions for the heliosphere that are consistent with local ISM parameters derived from heliosphere models. Since the upper size cutoff for the standard MRN size distribution is well below the size of the largest grains observed (for silicate grains; graphite grains sizes up to 1 μm are allowed), we also plot an MRN-type size distribution but with higher lower and upper size limits (0.01–1.0 μm), and using the same n_H and dust-to-gas ratio for normalization, that we refer to as a "shifted MRN" size distribution. We note that more recent detailed ISM grain models such as Zubko et al. (2004) have grain size distributions that differ substantially from a simple power law and in some cases extend significantly above the 0.25 μm cutoff of the MRN model. However, no proposed ISM grain model has a size distribution similar to that observed in the heliosphere (Draine 2009), because such a size distribution is inconsistent with observed extinction curves. We note, however, that the extinction curves that such grain models aim to match are derived from observations over kiloparsec-long lines of sight. This implies that either the size distribution in the LIC is very atypical of the ISM or it has been substantially distorted by the transit of the grains through the heliosphere. Our results provide insights into the effects of those distortions.

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If we assume that the shifted MRN distribution represents the grain size distribution in the undisturbed ISM, then, using the focusing SWMF polarity heliosphere model, we find that we would expect the detections shown as the blue points in Figure 14. For these points, we use the density results for the voxels consistent with the orbit of Ulysses, including ranges in x, y, and z of −3 to +3 AU, −1.3 to +5.4 AU, and −2.1 to +2.1 AU, respectively (using the coordinate system described in Section 2.1 above). The points are generally well above the actual detections except for the point at 0.0562 μm, for which no counts are found in the model for the region close to the Sun (see discussion below). This suggests that the focusing model provides too much focusing (at least for the assumed size distribution), as would be expected if the grains experienced a de-focusing phase as well as focusing during their transit of the heliosphere. If instead we use the actual observations (interpolated to get the grain densities at the calculated grain sizes), then we can infer the size distribution in the ISM by using the model enhancement/depletion of grains caused interaction with the heliosphere. We have done that, using spline interpolation on the data, with the results plotted as the red points in the figure. These points are correspondingly well below the shifted MRN model line (again with the exception of the 0.0562 μm grains). In going from the observed points to the size distribution in the undisturbed ISM, we divide by the enhancement factor, i.e., ratio of dust density for grains of a given size to that in the undisturbed ISM. Therefore, a larger enhancement leads to a lower value inferred for the ISM. Thus, the model with the focusing SWMF polarity appears to overpredict the focusing and predicts a very small dust-to-gas ratio in the LIC.

At the low end of the size distribution we have done a linear (in the log, i.e., power law) extrapolation of the data down from the ∼0.1 μm detection threshold of Ulysses to 0.01 μm to explore the implications for the grain size distribution. If the grain size distribution near the Sun really does extend in this way to smaller sizes, then the model results imply a flattening of the size distribution in the ISM for these small grain sizes. Since there is no data on interstellar grains of those sizes in the heliosphere, this result (which also only applies for the focusing-only SWMF) is purely speculative at this point.

If we instead interpret the dust measurements using the de-focusing SWMF polarity model the results are starkly different. Since all but the three largest grain sizes modeled (0.316, 0.562, and 1.0 μm) produce no counts for the voxels near the Sun, the inferred distribution of grains in the ISM has, except for grain sizes ⩾0.316 μm, very high lower limits that generally exceed the extended MRN distribution. Such a high density of dust would demand a small gas-to-dust mass ratio, R_g/d < 38, and would conflict with limits on total cosmic abundances for the elements that make up the dust. On the other hand, the results for the focusing SWMF polarity model (discussed above) implies a rather small dust density and thus a high gas-to-dust ratio, R_g/d ∼ 820 (ignoring the 0.0562 μm grain size). This would indicate a high degree of dust destruction in the ISM and only a small mass of heavy elements tied up in the dust. This again conflicts with observations, in particular those that indicate significant gas-phase depletion of several elements, e.g., Fe, Mg, and Si, from the gas phase (Slavin & Frisch 2008).

The fact that neither the focusing SWMF nor de-focusing SWMF models can be comfortably accommodated with our information on gas-phase elemental abundances in the LIC points to the limitations of constant polarity models for the SWMF. This is not surprising since the grains require ∼20 years to travel between the termination shock and inner heliosphere. The fact that the predictions of our two heliosphere models lead to bracketing of the likely possibilities for the inferred gas-to-dust ratio and grain size distribution suggests that grain trajectory modeling needs to include the time variation of the solar wind, and its magnetic field polarity in particular, over the solar cycle. The strong dependence of the grain density on the SWMF polarity shows that the recovery of the mass distribution of interstellar dust from in situ measurements requires a complete understanding of the effect of the solar cycle on the global heliosphere, and grain trajectory models that incorporate interactions with the 3D global heliosphere.

We note that the grains with the particular size of 0.0562 μm are predicted to have very low density (zero counts calculated) near the Sun for both heliosphere models, though for the focusing SWMF model a substantial amount of dust penetrates the heliopause. As discussed in Section 4 above, this appears to be caused by the size of the gyroradius for grains of this size, which is large enough that the grains decouple from the gas enough to penetrate the heliopause and termination shock, but small enough that their path is substantially diverted from its initial direction. We expect that additional perturbations to the initial conditions for the grains and especially spatial and temporal variations in the solar wind will wash out this feature leading to a relatively featureless grain size distribution such as is observed.

The heliosphere models that we use in this paper are consistent with all the information available up until recently on the ISM and the heliosphere. This includes the speed of the inflowing ISM and the orientation of the ISMF required to produce the asymmetry in the shape of the termination shock as revealed by the location of the Voyager 1 and Voyager 2 crossings. Recent data from IBEX (Möbius et al. 2012) indicate that the speed of the interstellar inflow is about 12% less than previously determined, ∼23 km s⁻¹ rather than 26.4 km s⁻¹. This lower velocity appears to rule out a bow shock ahead of the heliosphere (McComas et al. 2012). If this result holds up, then other parameters of the heliosphere models, e.g., ISMF strength, will likely require some small adjustments as well. We do not expect this to make any difference regarding the fundamental conclusions of this paper, especially since the models that we have used already do not have a shock transition between the ISM and the outer heliosheath.